Hostname: page-component-546b4f848f-zwmfq Total loading time: 0 Render date: 2023-05-31T19:19:37.202Z Has data issue: false Feature Flags: { "useRatesEcommerce": true } hasContentIssue false
  • English

On the Unilateral Contact Between Membranes. Part 1: Finite Element Discretization andMixed Reformulation

Published online by Cambridge University Press:  27 January 2009

F. Ben Belgacem ,
C. Bernardi ,
A. Blouza  and
M. Vohralík
F. Ben Belgacem
Affiliation:
L.M.A.C. (E.A. 2222), Département de Génie Informatique, Université de Technologie de Compiègne, Centre de Recherches de Royallieu, B.P. 20529, 60205 Compiègne Cedex, France
C. Bernardi*
Affiliation:
Laboratoire Jacques-Louis Lions, C.N.R.S. & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France
A. Blouza
Affiliation:
Laboratoire de Mathématiques Raphaël Salem (U.M.R. 6085 C.N.R.S.), Université de Rouen, avenue de l'Université, B.P. 12, 76801 Saint-Étienne-du-Rouvray, France
M. Vohralík
Affiliation:
Laboratoire Jacques-Louis Lions, C.N.R.S. & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France
*
bernardi@ann.jussieu.fr
Get access

Abstract

The contact between two membranes can be described by a system of variational inequalities, where the unknowns are the displacements of the membranes and the action of a membrane on the other one. We first perform the analysis of this system. We then propose a discretization, where the displacements are approximated by standard finite elements and the action by a local postprocessing. Such a discretization admits an equivalent mixed reformulation. We prove the well-posedness of the discrete problem and establish optimal a priori error estimates.

Keywords

unilateral contactelastic membranesvariational inequalities
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 1: Modelling and numerical methods in contact mechanics , 2009 , pp. 21 - 43
Copyright
© EDP Sciences, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bank, R.E., Rose, D.J.. Some error estimates for the box method. SIAM J. Numer. Anal., 24 (1987), 777–787. CrossRef
F. Ben Belgacem, C. Bernardi, A. Blouza, M. Vohralík. A finite element discretization of the contact between two membranes. Math. Model. Numer. Anal. (2009) DOI: 10.1051/m2an:2008041.
C. Bernardi, Y. Maday, F. Rapetti. Discrétisations variationnelles de problèmes aux limites elliptiques. Collection “Mathématiques et Applications" 45, Springer-Verlag, Berlin, 2004.
Brezis, H., Stampacchia, G.. Sur la régularité de la solution d'inéquations elliptiques. Bull. Soc. Math. France, 96 (1968), 153–180. CrossRef
F. Brezzi, W.W. Hager, P.-A. Raviart. Error estimates for the finite element solution of variational inequalities. II. Mixed methods. Numer. Math., 31 (1978/79), 1–16.
P.G. Ciarlet. The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics 40, Society for Industrial and Applied Mathematics, 2002.
Falk, R.S.. Error estimates for the approximation of a class of variational inequalities. Math. Comput., 28 (1974), 963–971. CrossRef
P. Grisvard. Elliptic Problems in Nonsmooth Domains. Pitman, 1985.
J. Haslinger, I. Hlaváček, J. Nečas. Numerical methods for unilateral problems in solid mechanics. In: Handbook of Numerical Analysis, Vol. IV, P.G. Ciarlet & J.-L. Lions eds. North-Holland, Amsterdam (1996), pp. 313–485.
N. Kikuchi, J. T. Oden. Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. Studies in Applied and Numerical Mathematics, Society for Industrial and Applied Mathematics, 1988.
Lions, J.-L., Stampacchia, G.. Variational inequalities. Comm. Pure and Appl. Math., 20 (1967), 493–519. CrossRef
P.-A. Raviart, J.-M. Thomas. A mixed finite element method for second order elliptic problems. In: Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics, 606, Springer, 1977, pp. 292–315.
Slimane, L., Bendali, A., Laborde, P.. Mixed formulations for a class of variational inequalities. Math. Model. Numer. Anal., 38 (2004), 177–201. CrossRef
Vohralík, M.. A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization. C. R. Math. Acad. Sci. Paris, 346 (2008), 687–690. CrossRef